### Upcoming Talks

#### Strukturtheorie-Seminar

**Title:**On the effective impedance of networks

**Speaker:**Anna Muranova (Univ. Bielefeld)

**Date:**7.11.2019, 11:00 s.t.

**Room:**Seminar room AE02, Steyrerg. 30/ground floor

**Abstract:**

Electrical networks with resistors can be considered as weighted graphs. Similarly, infinite electrical networks with resistors can be introduced. In this case the effective resistance of a network can be defined, and it is related to Laplace operator and random walk on graphs.

A natural generalization of a network with resistors is given by an electrical network with resistors, capacitors, and inductors (so-called network with impedances). It is more convenient for us to work with the admittance (the inverse of impedance). We define the mathematical notion of effective admittance $P$ of a finite network and consider it as a rational function of $\lambda$. Although initially $\lambda = i\omega$, where $\omega$ is the frequency of an alternating current, we consider more generally $\lambda$ as taking arbitrary complex values. We present some estimates of the function $P(\lambda)$. Moreover, we discuss the possibility to define an effective admittance of an infinite electrical network. The idea is to exhaust an infinite network by a sequence of finite networks.

#### Mathematisches Kolloquium

**Title:**Cichoń's Diagram and the cardinality of the continuum

**Speaker:**Prof. Dr. Martin Goldstern (TU Wien)

**Date:**25.10.2019, 14:00 (13:30 Buffet vor dem HS)

**Room:**HS BE01, Steyrergasse 30, EG

**Abstract:**

Cichoń's Diagram describes a partial order between 10 uncountable cardinals, among them cov(null), the smallest number of Lebesgue null sets needed to cover all real numbers, or non(meager), the smallest cardinality of a non-meager set (=set of second category), as well as aleph1 (smallest uncountable cardinal) and c (the cardinality of R, the set of all real numbers).

In 1963, Paul Cohen invented the method of "forcing" and used it to show the unprovability of Cantor's Continuum Hypothesis, or in other

words:

that the continuum does not necessarily have cardinality aleph1.

It is still open which values the other cardinal's in Cichoń's Diagram may take, but in a recent paper (with Jakob Kellner and Saharon Shelah) we could for the first time construct a set-theoretic universe in which all cardinals in Cichoń's Diagram have different values.

I will talk a bit about the cardinals in Cichoń's Diagram and hint at the methods which allow us to control/manipulate their values.

#### Strukturtheorie-Seminar

**Title:**Self-adjoint and Markovian extensions of quantum graphs

**Speaker:**Aleksey Kostenko (Universitäten Wien und Ljubljana)

**Date:**Donnserstag, 24.10.2019, 11 Uhr c.t.

**Room:**SR AE02, Steyrergasse 30, EG

**Abstract:**

The main focus is on the relationship between graph ends of an infinite metric graph and the space of self-adjoint extensions of the corresponding minimal Kirchhoff Laplacian. First, we introduce the notion of finite volume for (topological) ends of a metric graph and then investigate their relationship

with the deficiency indices of the Kirchhoff Laplacian. Moreover, it turns out that finite volume graph ends play a crucial role in the study of Markovian extensions. In particular, in the case of finitely many finite volume ends we are even able to provide a complete description of all Markovian extensions.

Based on joint work with D. Mugnolo (Hagen) and N. Nicolussi (Wien - Potsdam)